f is continuous in [a, b] and differentiable in (a, b) (where a>0 ) such that `f(a)/a=f(b)/b`. Prove that there exist `x_0 in (a, b)` such that `f'(x_0 ) = f(x_0)/x_0`

Let f(x)be continuous on [a,b], differentiable in (a,b) and f(x)ne0"for all"x in[a,b]. Then prove that there exists one c in(a,b)"such that"(f'(c))/(f(c))=(1)/(a-c)+(1)/(b-c).

Let f be continuous on [a,b],a>0, and differentiable on (a,b). Prove that there exists c in(a,b) such that (bf(a,b))/(b-a)=f(c)-cf'(c)

Let f be any continuously differentiable function on [a, b] and twice differentiable on (a, b) such that f(a)=f'(a)=0" and "f(b)=0 . Then-

Let f be arry continuously differentiable function on [a,b] and twice differentiable on (a.b) such that f(a)=f(a)=0 and f(b)=0. Then

Let f be continuous on [a, b] and assume the second derivative f" exists on (a, b). Suppose that the graph of f and the line segment joining the point (a,f(a)) and (b,f(b)) intersect at a point (x_0,f(x_0)) where a < x_0 < b. Show that there exists a point c in(a,b) such that f"(c)=0.

If f is continuous on [a;b] and f(a)!=f(b) then for any value c in(f(a);f(b)) there is at least one number x_(0) in (a;b) for which f(x_(0))=c

Let f be continuous on [a;b] and differentiable on (a;b) If f(x) is strictly increasing on (a;b) then f'(x)>0 for all x in(a;b) and if f(x) is strictly decreasing on (a;b) then f'(x)<0 for all x in(a;b)